super-Valdivia compact

Definition

• Compact Hausdorff space X is a super-Valdivia compact space if each x \in X is contained in some dense Σ-subset of X

Property

• Let f:X→Y be a continuous open surjection between compact Hausdorff spaces. Suppose, moreover, that Y has a dense set of G_δ points. If X is super-Valdivia , then Y is Corson.
• Let X and Y be nonempty compact Hausdorff spaces such that X has a dense set of G_δ points and X×Y is super-Valdivia compact, then X is Corson and Y super-Valdivia.
• Let X_a, a \in Λ be an arbitrary family of nonempty compact Hausdorff spaces such that each X_a has a dense subset of G_δ points. Then the follwing two conditions are equivalent.
  1. Π_{a \in Λ} X_a is super-Valdivia compact.
  2. X_a is a Corson compact for every a \in Λ.

Reference

Kalenda, Ondřej(CZ-KARLMP-MA),A characterization of Valdivia compact spaces, (English summary)Collect. Math. 51 (2000), no. 1, 59--81.